Prompted by the comments to a recent answer by YCor to a related question ([here][1]), I'd like to ask the following:

> **Q.** Does every cancellative duo semigroup embed into a group?

A (multiplicatively written) semigroup $S$ is cancellative if, for every $a \in S$, right and left multiplication by $a$ are both injective functions on $S$; and it is duo if $aS = Sa$ for all $a\in S$.

By a famous example of Mal'cev [Math. Ann. 113 (1937), No. 1, 686-691], not every cancellative semigroup is embedable into a group; so, one could preliminarily ask if Mal'cev's example is duo. (EDIT: it follows from [Pace Nielsen's answer below][2] that Mal'cev's example is *not* duo.) On the other hand, it is folklore (and easy to prove) that a _commutative_ semigroup embeds into a group if and only if it is cancellative. Duo semigroups lie halfway between the commutative and non-commutative worlds, and it seems therefore natural to ask whether they are closer to one or the other when it comes to embeddability.


  [1]: https://mathoverflow.net/a/480278/16537
  [2]: https://mathoverflow.net/a/480395/16537